You are mixing up several different computers here.

> Maybe they meant the ZX Spectrum II

No. There was never never a "Spectrum II".

The second model after the original 16K and 48K was the ZX Spectrum Plus, in a keyboard derived from the 1984 Sinclair QL design.

http://www.retro8bitcomputers.co.uk/Sinclair/ZXSpectrumPlus

Then the 3rd model was the ZX Spectrum 128, in the same keyboard, but with more ports and a large external heatsink.

https://www.computinghistory.org.uk/det/2584/sinclair-zx-spe...

> known to some as “The Toaster” for some reason.

Nope. The 128 was known as the "toastrack" for the heatsink.

> Rubber keyboard, I heard it referred to as dead-flesh.

Not since the Plus model, no.

After the 128, Amstrad bought the brand. It launched the ZX Spectrum +2 and +3.

https://www.computinghistory.org.uk/det/3648/Sinclair-ZX-Spe...

https://www.computinghistory.org.uk/det/509/Sinclair-ZX-Spec...

Those are the closest to the nonexistent model number "II" but they do not have the QL-derived keyboard.